let V be Z_Module; :: thesis: for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,INT
for i being Element of NAT st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v
let v be VECTOR of V; :: thesis: for F being FinSequence of V
for f being Function of the carrier of V,INT
for i being Element of NAT st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v
let F be FinSequence of V; :: thesis: for f being Function of the carrier of V,INT
for i being Element of NAT st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v
let f be Function of the carrier of V,INT; :: thesis: for i being Element of NAT st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v
let i be Element of NAT ; :: thesis: ( i in dom F & v = F . i implies (f (#) F) . i = (f . v) * v )
assume that
A1: i in dom F and
A2: v = F . i ; :: thesis: (f (#) F) . i = (f . v) * v
A3: F /. i = F . i by A1, PARTFUN1:def 6;
len (f (#) F) = len F by Def22;
then i in dom (f (#) F) by A1, FINSEQ_3:29;
hence (f (#) F) . i = (f . v) * v by A2, A3, Def22; :: thesis: verum